Who among the following constitutional authorities is responsible for administering the oath of office and secrecy to a State's Chief Minister?

The correct option is Governor of the State.

Step-by-step Explanation:
1. Constitutional Provision: According to Article 164(3) of the Constitution of India, before a Chief Minister or any State Minister enters upon their office, the Governor of the state administers the oath of office and secrecy to them according to the forms set out in the Third Schedule of the Constitution.
2. Executive Authority: The Governor serves as the constitutional head of the State executive. As part of their executive duties, the Governor appoints the Chief Minister and, on the advice of the Chief Minister, appoints other ministers, administering the oath of office and secrecy to all of them.
3. Distinction with Other Offices:
- The President of India administers the oath to the Prime Minister and Union Cabinet Ministers.
- The Chief Justice of India administers the oath to the President of India.
- The Speaker of the Legislative Assembly is elected by the members of the Assembly to preside over its legislative proceedings.